Theorem 5 ([1]}) 

If the regularized direction \(u^{\sf \tiny r}_{p}\)  with respect to the \(\ell _2\) -norm exists, then the gradient descent iterates with sufficiently small step size converge to the regularized direction \(u^{\sf \tiny r}_{2}\) , i.e., \(\lim _{t\rightarrow \infty } \frac{w_t}{\left\Vert w_t \right\Vert _2} = u^{\sf \tiny r}_{2}\) .

